Calculus 1120, Fall 2012 Dan Barbasch October 25, 2012 Dan - - PowerPoint PPT Presentation

Calculus 1120, Fall 2012

Dan Barbasch October 25, 2012

Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 1 / 1

Summary of material for Prelim II

Integration, chapter 8

1

Techniques of Integration 8.1-8.5

2

Numerical Integration 8.6

3

Improper Integrals 8.7

review problems are on page 507 Differential equations, 7.2, most of 9.1-9.4

1

Graphic methods, Slope fields 9.1, parts of 9.4 draw slope fields and some solutions for problems on page 540

2

Separable and Linear Equations 7.2, 9.3 page 433 1-8, 9-22, page 526 1-21

3

Applications, parts of 9.1-9.4. page 434 23-46, page 540 13-18

Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 2 / 1

Integration

Make sure you know the standard form of writing P(x)

Q(x) as a partial

fraction The standard form is a combination of x (±a2 + x2)n , 1 (±a2 + x2)n x (a2 − x2)n , 1 (a2 − x2)n In the first column substitute u = ±a2 + x2, u = a2 − x2. In the second column do a trigonometric substitution. The outcome is a trigonometric integral which you can do by double angle formulas, recursion formulas and so on.

Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 3 / 1

Numerical Integration

Make sure you review the left/right endoint approximations, and the

• ver/under estimates.

Formulas for error estimates will be given.

Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 4 / 1

Improper Integrals

You must divide the integral into a sum so that each integral has one

• singularity. Analyze the function at the singularity.

ln x << xa, (any a > 0) xa << bx, (any a > 0, b > 1) b−x << 1 xa , (any b > 1, a > 0) 1 xa << 1 ln x (any a > 0). as x → ∞. Remember that you must find a function g(x) to compare with, but also compute an integral to carry out the test. Often g(x) is 1/xp where you know the answer.

Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 5 / 1

Improper Integrals

An exception might be ∞

2 1 x(ln x)p dx.

A

2

1 x(ln x)p dx = ln A

ln 2

1 up du. lim

A→∞

A

2

1 x(ln x)p dx = lim

A→∞

ln A

ln 2

1 up du = lim

B→∞

B

ln 2

du up . So ∞

2

1 x(ln x)p dx = ∞

ln 2

1 up du. Contrast this with ∞

2 dx (ln x)p . In this case

0 ≤ 1 xap ≤ 1 (ln x)p and so the integral diverges for any p.

Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 5 / 1

Differential Equations

Problem 16, page 534 Carbon monoxide pollution. An executive conference room of a corporation contains 4500ft3 of air initially free of carbon monoxide. Starting at time t = 0, cigarette smoke containing 4% carbon monoxide is blown into the room at the rate of 0.3ft3/min. A ceiling fan keeps the air in the room well circulated and the air leaves the room at the same rate of 0.3ft3/min. Find the time when the concentration of carbon monoxide in the room reaches 0.01%.

Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 6 / 1

Differential Equations

Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 6 / 1

Differential Equations

Autonomous Equation y′ = F(y). Equilibrium Solution y = c where F(c) = 0. Problem: Consider the differential equation y′ = 2xy2. (i) Draw the direction field. Mark clearly where the slopes are negative, positive and zero. (ii) Find the equilibrium solution if there is one. Draw several solutions, in particular the ones with initial condition y(0) = 1 and y(0) = −1. (iii) What is the long term behaviour of the solutions (i.e. what is limx→∞ y(x))? (iv) Solve the differential equation. Compute the exact value of y(0.3) for the solution satisfying the initial condition y(0) = 1.

Dan Barbasch () Calculus 1120, Fall 2012 October 25, 2012 6 / 1